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SUNDAY SCIENCE: HIGH SCHOOL STUDENTS WHO CAME UP WITH
‘IMPOSSIBLE’ PROOF OF PYTHAGOREAN THEOREM DISCOVER 9 MORE
SOLUTIONS TO THE PROBLEM
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Sascha Pare
October 27, 2024
Live Science
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_ In a new peer-reviewed study, Ne'Kiya Jackson and Calcea Johnson
outlined 10 ways to solve the Pythagorean theorem using trigonometry,
including a proof they discovered in high school. _
Ne'Kiya Jackson and Calcea Johnson came up with an "impossible" proof
to the Pythagorean theorem when they were high school seniors., Calcea
Johnson
Two students who discovered a seemingly impossible proof to the
Pythagorean theorem in 2022 have wowed the math community again with
nine completely new solutions to the problem.
While still in high school, Ne'Kiya Jackson and Calcea Johnson from
Louisiana used trigonometry to prove the 2,000-year-old Pythagorean
theorem
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which states that the sum of the squares of a right triangle's two
shorter sides are equal to the square of the triangle's longest side
(the hypotenuse). Mathematicians had long thought that using
trigonometry to prove the theorem was unworkable, given that the
fundamental formulas for trigonometry are based on the assumption that
the theorem is true.
Jackson and Johnson came up with their "impossible" proof in answer to
a bonus question in a school math contest. They presented their work
at an American Mathematical Society meeting in 2023, but the proof
hadn't been thoroughly scrutinized at that point. Now, a new paper
published Monday (Oct. 28) in the journal American Mathematical
Monthly
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their solution held up to peer review. Not only that, but the two
students also outlined nine more proofs to the Pythagorean theorem
using trigonometry.
"To have a paper published at such a young age — it's really
mind-blowing," Johnson, who is now studying environmental engineering
at Louisiana State University, said in a statement emailed to Live
Science. "I am very proud that we are both able to be such a positive
influence in showing that young women and women of color can do these
things."
By proving Pythagoras [[link removed]]'
theorem using trigonometry, but without using the theorem itself, the
two young women overcame a failure of logic known as circular
reasoning. Trigonometry is a branch of mathematics
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how the sides, lengths and angles in a triangle are related, and as
such, the discipline often includes expressions of the Pythagorean
theorem. But Jackson and Johnson managed to prove the theorem using a
result of trigonometry called the Law of Sines, dodging circular
reasoning.
In the new study, and on top of their initial proof, the young
mathematicians described four new ways to prove Pythagoras' theorem
using trigonometry, as well as a novel method that revealed five more
proofs, totaling 10 proofs.
Jackson and Johnson are only the third and fourth people known to have
proven the Pythagorean theorem using trigonometry and without
resorting to circular reasoning. The two other people were
professional mathematicians, according to the statement.
"I didn't think it would go this far," Jackson, who currently studies
pharmacology at the Xavier University of Louisiana, said in the
statement. "I was pretty surprised to be published."
In the paper, Jackson and Johnson say there are two ways to present
trigonometry and its functions sine and cosine, but these versions are
often conflated into one. Sine and cosine are ratios that are defined
in the context of a triangle's right angle, and they can be presented
according to either the trigonometric method or a method that uses
polynomials of complex numbers, according to the paper.
The conflation means that "trying to make sense of trigonometry can be
like trying to make sense of a picture where two different images have
been printed on top of each other," Jackson and Johnson wrote.
By teasing the two methods apart, researchers can discover "a large
collection of new proofs of the Pythagorean theorem," the young
mathematicians added.
_SASCHA PARE is a U.K.-based trainee staff writer at Live Science. She
holds a bachelor’s degree in biology from the University of
Southampton in England and a master’s degree in science
communication from Imperial College London. Her work has appeared in
The Guardian and the health website Zoe. Besides writing, she enjoys
playing tennis, bread-making and browsing second-hand shops for hidden
gems._
_LIVE SCIENCE is one of the biggest and most trusted popular science
websites operating today, reporting on the latest discoveries,
groundbreaking research and fascinating breakthroughs that impact you
and the wider world._
_We believe that science can help explain the things that matter to
you and shine a light on everything from the mysteries of our universe
to the inner workings of an atom. Our team of experienced editors and
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stories with clarity, authority and humor._
_Whether you’re interested in dinosaurs or archaeology, weird
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begins here._
_Subscribe to Live Science
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__
It Might Be Possible to Detect Gravitons After All
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by Charlie Wood
Quanta Magazine
October 30, 2024
A new experimental proposal suggests detecting a particle of gravity
is far easier than anyone imagined. Now physicists are debating what
it would really prove.
* Science
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* mathematics
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* Pythagorean theorem
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* high school students
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